Head off into the nebulae and vast open regions (ie. those things at the end or side of blocks, like Delta Volanis or the Zenas Expanse). You can pick up a couple daily missions to explore 3 systems & earn dilithium for it.

The missions are a real mix, some use standard races, others are purely randomized. You might get a space investigation mission, or a ground "scan X things" mission, or a "explore these caves / outpost." Oh, and of course, combat missions. It's really the closest thing to exploring strange new worlds in-game.